-
Measurement
Base quantities and
their units; mass (kg), length (m), time (s),
current (A),
temperature (K), amount of
substance (mol)
:
Derived units as products or quotients
of the base units:
Prefixes and their symbols
to indicate decimal sub-multiples or multiples of
both base and derived units:
Estimates of physical
quantities:
When making an estimate, it
is only reasonable to give the figure to 1 or at
most 2 significant figures since an
estimate is not very precise.
?
Occasionally, students are asked to
estimate the area under a graph.
The
usual method of counting squares within the
enclosed area is
used. (eg. Topic 3
(Dynamics), N94P2Q1c)
Often, when making an estimate, a
formula and a simple calculation
may be
involved.
?
EXAMPLE 1:
Estimate the
average running speed of a typical 17-year-
old
?
s 2.4-km run.
velocity = distance / time = 2400 /
(12.5 x 60) = 3.2 ≈3 ms
-1
EXAMPLE 2:
Which estimate is
realistic?
Distinction
between systematic errors (including zero errors)
and random
errors and between precision
and accuracy:
Random error:
is the type of error which causes readings to
scatter about
the true value.
Systematic error:
is the
type of error which causes readings to deviate in
one
direction from the true value.
Precision:
refers to the
degree of agreement (scatter, spread) of
repeated
measurements of the same quantity. {NB:
regardless of whether or not they
are
correct.}
Accuracy:
refers
to the degree of agreement between the result of a
measurement and the true value of the
quantity.
Assess the
uncertainty in a derived quantity by simple
addition of actual,
fractional or
percentage uncertainties (a rigorous statistical
treatment is not
required).
For a quantity x = (2.0 ± 0.1)
mm,
Actual/ Absolute
uncertainty, Δ x = ± 0.1 mm
Fractional uncertainty, Δxx =
0.05
100% = 5
%
?
Percentage
unce
rtainty, Δxx
If p = (2x + y) / 3 or p = (2x -
y) / 3 , Δp = (2Δx + Δy) / 3
If r = 2xy
3
or r
= 2x / y
3
, Δr /
r = Δx / x + 3Δy / y
Actual
error must be recorded to only
1
significant figure
, &
The
number of
decimal places
a
calculated quantity should have is
determined by its actual error.
For eg, suppose g has been initially
calculated to be 9.80645
ms
-2
& Δg
has been initially calculated to be
0.04848 ms
-2
. The final
value of Δg must
be recorded as 0.05
ms
-2
{
1 sf }, and
the appropriate recording of g is
(9.81
± 0.05) ms
-2
.
Distinction between scalar and vector
quantities
:
Representation of vector as two
perpendicular components:
In the
diagram below, XY represents a flat kite of weight
4.0 N. At a certain
instant, XY is
inclined at 30° to the horizontal
and
the wind exerts a steady
force of 6.0 N
at right angles to XY so that the kite flies
freely.
Kinematics
Displacement, speed, velocity and
acceleration:
Distance:
Total length covered irrespective of the direction
of motion.
Displacement:
Distance moved in a certain direction.
Speed:
Distance travelled
per unit time.
Velocity:
is
defined as the rate of change of displacement, or,
displacement
per unit time
{
NOT
:
displacement
over
time, nor,
displacement
per second
,
nor, rate of
change of displacement per
unit time}
Acceleration:
is
defined as the rate of change of velocity.
Using graphs to find
displacement, velocity and acceleration:
?
?
The area under a velocity-time graph is
the change in displacement.
The gradient of a displacement-time
graph is the {instantaneous}
velocity.
The gradient of a velocity-
time graph is the acceleration.
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The 'SUVAT'
Equations of Motion
The most important
word for this chapter is SUVAT, which stands for:
?
?
?
?
?
S (displacement),
U (initial velocity),
V (final velocity),
A (acceleration) and
T (time)
of a
particle that is in motion.
Below is a list of the equations you
MUST memorise, even if they are in the
formula book, memorise them anyway, to
ensure you can implement them
quickly.
1.
v = u +at
2.
s = ? (u + v)
t
3.
v
2
=
u
2
+ 2as
4.
s = ut +
?at
2
derived from
definition of acceleration: a = (v
–
u) / t
derived
from the area under the v-t graph
derived from equations (1) and (2)
derived from equations (1) and (2)
These equations apply only if the
motion takes place along a straight line
and the acceleration is constant;
{hence, for eg., air resistance must be
negligible.}
Motion of bodies falling in
a uniform gravitational field with air resistance:
Consider a body moving in a uniform
gravitational field under 2 different
conditions:
Without Air
Resistance:
Assuming
negligible air resistance
, whether the
body is moving up, or at the
highest
point or moving down, the weight of the body, W,
is the only force
acting on it, causing
it to experience a constant acceleration. Thus,
the
gradient of the v-t graph is
constant throughout its rise and fall. The body is
said to undergo free fall.
With Air Resistance:
If air resistance is NOT
negligible
and if it is projected
upwards with the
same initial velocity,
as the body moves upwards,
both
air resistance and
weight act
downwards
. Thus its speed
will decrease at a rate greater than
9.81 ms
-2
. This
causes the
time taken to reach its
maximum height reached
to be lower than
in the case with no air resistance. The max height
reached is
also reduced.
At the highest point, the body is
momentarily at rest; air resistance
becomes zero and hence the only force
acting on it is the weight. The
acceleration is thus 9.81
ms
-2
at this point.
As a body falls, air resistance opposes
its weight. The downward
acceleration
is thus less than 9.81 ms
-2
.
As air resistance increases with
speed,
it eventually equals its weight (but in opposite
direction). From then
there will be no
resultant force acting on the body and it will
fall with a
constant speed, called the
terminal
velocity
.
Equations for the horizontal and
vertical motion:
Parabolic
Motion: tan θ = v
y
/
v
x
θ: direction
of tangential velocity {NOT: tan θ =
s
y
/
s
x
}
Dynamics
Newton's laws of motion:
Newton's First Law
Every body continues in a state of rest
or uniform motion in a straight line
unless a net (external) force acts on
it.
Newton's Second Law
The rate of change of momentum of a
body is directly proportional to the net
force acting on the body, and the
momentum change takes place in the
direction of the net force.
Newton's Third Law
When object X exerts a force on object
Y, object Y exerts a force
of the same
type
that is equal in
magnitude and opposite in direction on object X.
The two forces ALWAYS act on different
objects and they form an
action-
reaction pair
.
Linear momentum and its conservation: